Approximate Analytic Estimate of Axial Fluid Conduction in Laminar Forced Convection Tube Flows with Zero-to-Uniform Step Wall Heat Fluxes

The influence of axial fluid conduction on low Péclet number flows in the thermal entrance region of long circular tubes is investigated in this theoretical study. The convective heat transport of viscous fluids relates to a specific condition under which the first part of the tube ( x h 0) is insulated and the second part of the tube ( x > 0) receives a heat flux of uniform intensity. A conjugate one-dimensional lumped model produces a solution of compressed algebraic form that is able to deliver dependable mean bulk temperatures that are in perfect agreement with those obtained numerically by the formal conjugate two-dimensional distributed model. As a by-product of the succession of algebraic calculations within the platform of the lumped model, the critical Péclet number Pe cr has been easily quantified. This number is a figure-of-merit of remarkable importance in the modeling of forced heat convection in tubes because it establishes the threshold between two contrasting situations: one embracing axial fluid conduction (finite Pe) and the other implicating negligible axial fluid conduction (Pe M X ). In addition, the local wall temperatures were calculated with an approximate engineering procedure, showing good agreement with those determined numerically by the formal conjugate two-dimensional distributed model.     

The effect of wall conduction for the extended Graetz problem for laminar and turbulent channel flows

Axial heat conduction effects within the fluid can be important for duct flows if either the Prandtl number is relatively low (liquid metals) or if the dimensions of the duct are small (micro heat exchanger). In addition, axial heat conduction effects in the wall of the duct might be of importance. The present paper shows an entirely analytical solution to the extended Graetz problem including wall conduction (conjugate extended Graetz problem). The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is relatively simple to compute and valid for all Péclet numbers. The analytical results are compared to own numerical calculations with FLUENT and good agreement is found.     

Conjugated heat transfer in circular ducts with a power-law laminar convection fluid flow

This work deals with the study of the steady-state analysis of conjugated heat transfer process for the thermal entrance region of a developed laminar-forced convection flow of a power-law fluid in a circular tube. A known uniform heat flux is applied at the external surface of the tube. The energy equation in the fluid is solved analytically using the integral boundary layer approximation by neglecting the heat generation by viscous dissipation and the axial heat conduction in the fluid. This solution is coupled to the Laplace equation for the solid, where the axial heat conduction effects are taken into account. The governing equations are reduced to an integro-differential equation which is solved by analytical and numerical methods. The results are shown for different parameters such as conduction parameter, α, the aspect ratio of the tube, ε and the index of power-law fluid, n.     

Fully developed forced convection of the Phan-Thien-Tanner fluid in ducts with a constant wall temperature

Two problems of laminar-forced convection in pipes and channels, under fully developed conditions, are solved for an imposed constant temperature at the wall, with fluids obeying the simplified Phan-Thien-Tanner (SPTT) model. The fluid properties are taken as constants and axial conduction is negligible. The first case represents the asymptotic behaviour of the Graetz problem for the SPTT fluid, i.e., equilibrium between axial convection and radial conduction of thermal energy with negligible viscous dissipation. The solution is given by an analytical expression but it is only approximate (within 0.3%) as it was obtained with an algebraic method based on successive approximations. The second problem has an exact analytical solution representing the equilibrium between viscous dissipation and radial heat conduction, with negligible axial convection and a constant wall temperature.     

Thermal and heat transfer characteristics in a latent heat storage system using lauric acid

The thermal and heat transfer characteristics of lauric acid during the melting and solidification processes were determined experimentally in a vertical double pipe energy storage system. In this study, three important subjects were addressed. The first one is temperature distributions and temporal temperature variations in the radial and axial distances in the phase change material (PCM) during phase change processes. The second one is the thermal characteristics of the lauric acid, which include total melting and total solidification times, the nature of heat transfer in melted and solidified PCM and the effect of Reynolds and Stefan numbers as inlet heat transfer fluid (HTF) conditions on the phase transition parameters. The final one is to calculate the heat transfer coefficient and the heat flow rate and also discuss the role of Reynolds and Stefan numbers on the heat transfer parameters. The experimental results proved that the PCM melts and solidifies congruently, and the melting and solidification front moved from the outer wall of the HTF pipe (HTFP) to the inner wall of the PCM container in radial distances as the melting front moved from the top to the bottom of the PCM container in axial distances. However, it was difficult to establish the solidification proceeding at the axial distances in the PCM. Though natural convection in the liquid phase played a dominant role during the melting process due to buoyancy effects, the solidification process was controlled by conduction heat transfer, and it was slowed by the conduction thermal resistance through the solidified layer. The results also indicated that the average heat transfer coefficient and the heat flow rate were affected by varying the Reynolds and Stefan numbers more during the melting process than during the solidification process due to the natural convection effect during the melting process.     

Exploration of thermal Péclet number,vortex viscosity,and Reynolds number on two-dimensional flow of micropolar fluid through a channel due to mixed convection

The present theoretical investigation is conducted on a micropolar fluid medium channel in the presence of mixed and nonlinear convection with the assumptions of thermal radiation and species reactive agents. The nonlinear governing equations, which describe the micropolar fluid flow and energy, are converted into ordinary differential equations using appropriate similarity variables. With the Runge–Kutta–Fehlberg method, the resultant equations are numerically solved. The physical characteristics of flow restrictions over velocity, microrotation, energy, and concentration profile are plotted and discussed. Further, the impact of several dimensionless parameters on Nusselt and Sherwood numbers is investigated and depicted graphically. In addition to observing flow patterns, contour plots of streamlines are plotted and discussed. It is demonstrated that the dimensionless velocity, temperature, and concentration of micropolar fluid have a maximum value at the center of the channel. However, the microrotation velocity of the micropolar fluid has both maxima and minima. The thermal and solutal properties of micropolar fluid influence heat and mass transport rates, that is, mixed convection and buoyancy parameter boost up the local heat transfer at the surface. Finally, Péclet number and chemically reactive parameters boost up the local mass transfer at the surface.     

Limitations of the lumped model for in‐tube laminar forced convection interacting with cross forced convection flows

The short communication addresses forced convection heat transfer of a viscous fluid flowing in a tube with fully developed laminar velocity and uniform entrance temperature that exchanges heat convection with a surrounding fluid. The idea is to implement the lumped model following the footsteps of the potent lumped model within unsteady heat conduction, instead of the differential model. The computed mean bulk temperatures of the viscous fluid expressed by a simple exponential function agrees well with the baseline numerical mean bulk temperatures when the internal fluid is air, water or crude oil and the external fluid is air for small values of the modified Biot numbers and maximum Reynolds numbers are satisfied.     

Axial dispersion in shell-and-tube heat exchangers

The effect of the deviation of the actual three-dimensional flow field on the shell-side from the frequently assumed one-dimensional uniform axial plug flow can be taken into account by superimposed axial dispersion in the fluid. The measure for axial dispersion is the Péclet number which can vary from infinite (no dispersion) to zero (complete axial mixing). For the fast and more reliable calculation of transient processes with the axial dispersion model, the Péclet number has to be known. A residence time distribution measurement technique for the determination of shell-side dispersive Péclet numbers is described and used to determine Péclet numbers for different shell-to-baffle clearances, numbers of baffles, and axial plug flow Reynolds numbers. Measurements with water reveal that Péclet numbers from 15 to 160 can occur and axial dispersion cannot be neglected in many cases.     

Numerical study of moving fin with thermal properties

In the present study, the influence of numerical analysis involves the transfer of heat for moving fin under convection and radiation with fluid properties depending on internal heat generation is considered. The temperature field for the fin is derived based on the defined geometry. This is transformed into a linear model as per the boundary conditions. The dimensionless analysis is performed, and the equations obtained are solved by Runge–Kutta Fehlberg fourth-order method coupled with the shooting method. The effects of the relative parameters are studied and depicted graphically. Finally, the computational observation indicates that there is an enhancement of temperature variation under the influence of thermal conductivity gradient, heat generation, and opposite behavior results. In understanding the thermal characteristics of various effects, it was seen that thermo geometric film, radiation conduction, and Peclet number were very useful for many industrial applications. To validate the numerical scheme, a comparative study has been done with an earlier result on a few nondimensional parameters and was found in good agreement.     

Transient heat transfer in a heat‐generating fin with radiation and convection with temperature‐dependent heat transfer coefficient

The transient heat transfer in a heat‐generating fin with simultaneous surface convection and radiation is studied numerically for a step change in base temperature. The convection heat transfer coefficient is assumed to be a power law function of the local temperature difference between the fin and its surrounding fluid. The values of the power exponent n are chosen to include simulation of natural convection (laminar and turbulent) and nucleate boiling among other convective heat transfer modes. The fin is assumed to have uniform internal heat generation. The transient response of the fin depends on the convection‐conduction parameter, radiation‐conduction parameter, heat generation parameter, power exponent, and the dimensionless sink temperature. The instantaneous heat transfer characteristics such as the base heat transfer, surface heat loss, and energy stored are reported for a range of values of these parameters. When the internal heat generation exceeds a threshold the fin acts as a heat sink instead of a heat source. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21012     

A numerical investigation into the charge behaviour of a spherical phase change material particle for high temperature thermal energy storage in packed beds

基于高温相变材料,对填充床储热系统中储热单元球体的储热性能进行了模拟研究.研究了不同传热流体温度和球体直径对球体储热性能的影响规律,对导热为主的相变储热过程与导热和自然对流共同作用的相变储热过程进行了比较分析,同时还探讨了高温辐射换热的影响.结果表明,相变时间随球体直径的增大而增大,随传热流体温度的增大而减小.当考虑相变区域自然对流时,总的相变时间显著减少,和单纯导热相比,完全相变时间缩短了近16%.在导热和自然对流的基础上加上辐射传热后可以看出,辐射换热强化了球体内的传热过程,加快了相变材料的熔化速度,强化了自然对流的作用.     

Transient Laminar MHD Natural Convection Cooling in a Vertical Cylinder

A numerical study is presented of transient laminar natural convection cooling of an electrically conductive fluid, placed in a vertical cylinder in the presence of an axial magnetic field. The cylindrical wall is suddenly cooled to a uniform temperature, thus setting the fluid to motion. The cooling process starts with the development of momentum and thermal boundary layers along the cylindrical cold wall, followed by the intrusion of the cooled fluid into the bulk, and finally, by fluid stratification. A range of Hartmann, Rayleigh, and Prandtl numbers are studied for which the flow remains laminar in all stages. It is found that the increase of the magnetic field reduces the heat transfer rate and decelerates the cooling process. This can be attributed to the damping of the fluid motion by the magnetic field, which results in the domination of conduction over convection heat transfer. The increase of the Rayleigh number enhances heat transfer, but the cooling process lasts longer due to the higher temperature of the hot fluid. The flow deceleration and the reduction of heat transfer are less intense for fluids with low Prandtl number.     

Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects

A modified Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Péclet number, Brinkman number).     

Effect of Radiative Heat Transfer on Three-Dimensional Double Diffusive Natural Convection

This work presents a numerical study of the effect of the radiative heat transfer on the three-dimensional double diffusive convection in a differentially heated cubic cavity for different optical parameters of the medium. This numerical study is conducted for fixed Prandtl, Rayleigh, and Lewis numbers, Pr = 13.6, Ra = 10⁵, Le = 2, and buoyancy ratio N in the range [–2, 0]. The natural convection equations, using the Boussinesq approximation for the treatment of buoyancy term in the momentum equation, are expressed using the vorticity–stream function formulation. These equations and the radiative transfer equation are discretized, respectively, with the control volume finite difference method and the FTn finite volume method. The influences of the optical thickness and the conduction–radiation parameter of the semitransparent fluid on heat and mass transfer are depicted. Results show different transitions of the structure of the main flow when varying the conduction–radiation parameter and the optical thickness.     

BUOYANCY-AIDED MIXED CONVECTION WITH CONDUCTION AND SURFACE RADIATION FROM A VERTICAL ELECTRONIC BOARD WITH A TRAVERSABLE DISCRETE HEAT SOURCE

This article presents the results of a comprehensive fundamental numerical study of the problem of buoyancy-aided mixed convection with conduction and surface radiation from a vertical electronic board provided with a traversable, flush-mounted, discrete heat source. Air, a radiatively transparent medium, is considered to be the cooling agent. The governing equations in primitive variables for fluid flow and heat transfer are first converted into stream function–vorticity form, and are later converted into algebraic form using the finite-volume method. The resulting finite-difference equations are solved by Gauss-Seidel iterative technique. The governing equation for temperature distribution along the electronic board is obtained by appropriate energy balance. The effects of pertinent parameters, viz., location of the discrete heat source, surface emissivity of the board, and modifiedRichardson number, on various results, including local temperature distribution along the board, maximum board temperature, and contributions of convection and surface radiation to heat dissipation from the board, are studied in great detail. The fact that any design calculation that ignores surface radiation in problems of this kind would be error-prone is clearly highlighted.     

Numerical study of fluid flow and heat transfer in microchannel cooling passages

Numerical investigation was conducted for fluid flow and heat transfer in microchannel cooling passages. Effects of viscosity and thermal conductivity variations on characteristics of fluid flow and heat transfer were taken into account in theoretical modeling. Two-dimensional simulation was performed for low Reynolds number flow of liquid water in a 100 μm single channel subjected to localized heat flux boundary conditions. The velocity field was highly coupled with temperature distribution and distorted through the variations of viscosity and thermal conductivity. The induced cross-flow velocity had a marked contribution to the convection. The heat transfer enhancement due to viscosity-variation was pronounced, though the axial conduction introduced by thermal-conductivity-variation was insignificant unless for the cases with very low Reynolds numbers.     

Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

Laminar flow and heat transfer in plate-fin triangular ducts in thermally developing entry region

Laminar forced flow and heat transfer in plate-fin isosceles triangular ducts encountered in compact heat exchangers is investigated. The flow is hydrodynamically fully developed, but developing thermally under uniform temperature conditions. Heat conduction in the fin of finite conductance and convection in the fluid are analyzed simultaneously as a conjugate problem. The study covers a wide range of apex angles from 30° to 120°, and fin conductance parameters from 0 to infinitely large. Nusselt numbers in the developing and fully developed regions for various apex angles and fin conductance parameters are obtained, which can be used in estimation of heat transfer characteristics in plate-fin compact heat exchangers with fins of various conductivities and thickness.     

Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation

Graetz problem inside the microtube is revisited considering rarefaction effect, viscous dissipation term and axial conduction in the fluid for uniform wall temperature boundary condition in the slip flow regime. The flow is assumed to be hydrodynamically fully developed, thermally developing, and the velocity profile is solved analytically. The temperature field is determined by the numerical solution of the energy equation. The rarefaction effect is imposed to the problem via velocity-slip and temperature jump boundary conditions. The local and fully developed Nu numbers are obtained in terms of dimensionless parameters; Pe, Kn, Br, κ. Fully developed Nu numbers and the thermal entrance length are found to increase by the presence of the finite axial conduction.     

Laminar conjugate mixed convection in a vertical channel with heat generating components

Conjugate mixed convection arising from protruding heat generating ribs attached to substrates (printed circuit boards) forming channel walls is numerically studied. The substrates with ribs form a series of vertical parallel plate channels. Assuming identical disposition and heat generation of the ribs on each board, a channel with periodic boundary conditions in the transverse direction is considered for analysis. The governing equations are discretised using a control volume approach on a staggered mesh and a pressure correction method is employed for the pressure–velocity coupling. The solid regions are considered as fluid regions with infinite viscosity and the thermal coupling between the solid and fluid regions is taken into account by the harmonic thermal conductivity method. Parametric studies are performed by varying the heat generation based Grashof number in the range 10⁴–10⁷ and the fan velocity based Reynolds number in the range 0–1500, with air as the working medium. Results are obtained for the velocity and temperature distributions, natural convection induced mass flow rate through the channel, the maximum temperatures in the heat sources and the local Nusselt numbers. The natural convection induced mass flow rate in mixed convection is correlated in terms of the Grashof and Reynolds numbers. In pure natural convection the induced mass flow rate varies as 0.44 power of Grashof number. The maximum dimensionless temperature is correlated in terms of pure natural convection and forced convection inlet velocity asymptotes. For the parameter values considered, the heat transferred to the working fluid via substrate heat conduction is found to account for 41–47% of the heat removal from the ribs.